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Engineering instructors often use physical manipulatives such as foam beams, rolling cylinders, and large representations of axis systems to demonstrate mechanics concepts and help students visualize systems. Additional benefits are possible when manipulatives are in the hands of individual students or small teams of students who can explore concepts at their own pace and focus on their specific points of confusion. Online learning modalities require new strategies to promote spatial visualization and kinesthetic learning. Potential solutions include creating videos of the activities, using CAD models to demonstrate the principles, programming computer simulations, and providing hands-on manipulatives to students for at-home use. This Work-in-Progress paper discusses our experiences with this last strategy in statics courses two western community colleges and a western four-year university where we supplied students with their own hands-on kits. We have previously reported on the successful implementation of a hands-on statics kit consisting of 3D printed components and standard hardware. The kit was originally designed for use by teams of students during class to engage with topics such as vectors, moments, and rigid body equilibrium. With the onset of the COVID-19 pandemic and shift to online instruction, the first author developed a scaled down version of themore »kit for at-home use by individual students and modified the associated activity worksheets accordingly. For the community college courses, local students picked up their models at the campus bookstore. We also shipped some of the kits to students who were unable to come to campus, including some in other countries. Due to problems with printing and availability of materials, only 18 kits were available for the class of 34 students at the university implementation. Due to this circumstance, students were placed in teams and asked to work together virtually, one student showing the kit to the other student as they worked through the worksheet prompts. One community college instructor took this approach as well for a limited number of international students who did not receive their kits in a timely manner due to shipping problems. Two instructors assigned the hands-on kits as asynchronous learning activities in their respective online courses, with limited guidance on their use. The third used the kits primarily in synchronous online class meetings. We found that students’ reaction to the models varied by pilot site and presume that implementation differences contributed to this variation. In all cases, student feedback was less positive than it has been for face-to-face courses that used the models from which the take home kit was adapted. Our main conclusion is that implementation matters. Doing hands-on learning in an online course requires some fundamental rethinking about how the learning is structured and scaffolded.« less

This NSF-IUSE exploration and design project began in fall 2018 and features cross-disciplinary collaboration between engineering, math, and psychology faculty to develop learning activities with 3D-printed models, build the theoretical basis for how they support learning, and assess their effectiveness in the classroom. We are exploring how such models can scaffold spatial skills and support learners’ development of conceptual understanding and representational competence in calculus and engineering statics. We are also exploring how to leverage the model-based activities to embed spatial skills training into these courses. The project’s original focus was on group learning in classroom activities with shared manipulatives. After a year of development and pilot activities, we commenced data collection in classroom implementations of a relatively mature curriculum starting fall 2019. Data collection ended abruptly in March 2020 when we had to shift gears in the context of a shift to online learning amid the COVID-19 pandemic. With uncertainty as to when the use of shared hands-on models in a collaborative in-person learning context would be feasible again, it was clear a change in approach would be necessary. We have since developed new versions of the models and associated curriculum designed for independent at-home use in the contextmore »of online learning. We implemented the new curricula in an online statics courses in fall 2020 and in multiple sections of online calculus courses in winter 2021. In this paper, we describe our strategies for implementing hands-on learning at home. We also present some example activities and compare the approach to the face-to-face versions. Finally, we compare student feedback results on the online activities to analogous feedback data from the classroom implementations and discuss implications for the anticipated return to face-to-face learning in the classroom.« less

A growing body of research indicates spatial visualization skills are important to success in many STEM disciplines, including several engineering majors that rely on a foundation in engineering mechanics. Many fundamental mechanics concepts such as free-body diagrams, moments, and vectors are inherently spatial in that application of the concept and related analytical techniques requires visualization and sketching. Visualization may also be important to mechanics learners’ ability to understand and employ common mechanics representations and conventions in communication and problem solving, a skill known as representational competence. In this paper, we present early research on how spatial abilities might factor in to students’ conceptual understanding of vectors and associated representational competence. We administered the Mental Cutting Test (MCT), a common assessment of spatial abilities, in the first and last week of the term. We also administered the Test of Representational Competence with Vectors (TRCV), a targeted assessment of vector concepts and representations, in week one and at mid-term. The vector post-test came after coverage of moments and cross products. We collected this assessment data in statics courses across multiple terms at three different colleges. To understand how spatial skills relate to the development of representational competence, we use a multiple regressionmore »model to predict TRCV scores using the pre-class MCT scores as well as other measures of student preparation in the form of grades in prerequisite math and physics coursework. We then extend the analysis to consider both MCT and TRCV scores as predictors for student performance on the Concept Assessment Test in Statics. We find that spatial abilities are a factor in students’ development of representational competence with vectors. We also find that representational competence with vectors likely mediates the importance of spatial abilities to student success in developing broader conceptual understanding in statics. We conclude by discussing implications for mechanics instruction.« less

The landscapes of many elementary, middle, and high school math classrooms have undergone major transformations over the last half-century, moving from drill-and-skill work to more conceptual reasoning and hands-on manipulative work. However, if you look at a college level calculus class you are likely to find the main difference is the professor now has a whiteboard marker in hand rather than a piece of chalk. It is possible that some student work may be done on the computer, but much of it contains the same type of repetitive skill building problems. This should seem strange given the advancements in technology that allow more freedom than ever to build connections between different representations of a concept. Several class activities have been developed using a combination of approaches, depending on the topic. Topics covered in the activities include Riemann Sums, Accumulation, Center of Mass, Volumes of Revolution (Discs, Washers, and Shells), and Volumes of Similar Cross-section. All activities use student note outlines that are either done in a whole group interactive-lecture approach, or in a group work inquiry-based approach. Some of the activities use interactive graphs designed on desmos.com and others use physical models that have been designed in OpenSCAD and 3D-printedmore »for students to use in class. Tactile objects were developed because they should provide an advantage to students by enabling them to physically interact with the concepts being taught, deepening their involvement with the material, and providing more stimuli for the brain to encode the learning experience. Web-based activities were developed because the topics involved needed substantial changes in graphical representations (i.e. limits with Riemann Sums). Assessment techniques for each topic include online homework, exams, and online concept questions with an explanation response area. These concept questions are intended to measure students’ ability to use multiple representations in order to answer the question, and are not generally computational in nature. Students are also given surveys to rate the overall activities as well as finer grained survey questions to try and elicit student thoughts on certain aspects of the models, websites, and activity sheets. We will report on student responses to the activity surveys, looking for common themes in students’ thoughts toward specific attributes of the activities. We will also compare relevant exam question responses and online concept question results, including common themes present or absent in student reasoning.« less

Perusal of any common statics textbook will reveal a reference table of standard supports in the section introducing rigid body equilibrium analysis. Most statics students eventually memorize a heuristic approach to drawing a free-body diagram based on applying the information in this table. First, identify the entry in the table that matches the schematic representation of a connection. Then draw the corresponding force and/or couple moment vectors on the isolated body according to their positive sign conventions. Multiple studies have noted how even high performing students tend to rely on this heuristic rather than conceptual reasoning. Many students struggle when faced with a new engineering connection that does not match an entry in the supports table. In this paper, we describe an inquiry-based approach to introducing support models and free-body diagrams of rigid bodies. In a series of collaborative learning activities, students practice reasoning through the force interactions at example connections such as a bolted flange or a hinge by considering how the support resists translation and rotation in each direction. Each team works with the aid of a physical model to analyze how changes in the applied loads affect the reaction components. A second model of the isolated bodymore »provides opportunity to develop a tactile feel for the reaction forces. We emphasize predicting the direction of each reaction component, rather than following a standard sign convention, to provide opportunities for students to practice conceptual application of equilibrium conditions. Students’ also draw detailed diagrams of the force interactions at the mating surfaces in the connection, including distributed loadings when appropriate. We use equivalent systems concepts to relate these detailed force diagrams to conventional reaction components. Targeted assessments explore whether the approach described above might improve learning outcomes and influence how students think about free-body diagrams. Students use an online tool to attempt two multiple-choice concept questions after each activity. The questions represent near and far transfer applications of the concepts emphasized and prompt students for written explanation. Our analysis of the students’ explanations indicates that most students engage in the conceptual reasoning we encourage, though reasoning errors are common. Analysis of final exam work and comparison to an earlier term in which we used a more conventional approach indicate a majority of students incorporate conceptual reasoning practice into their approach to free-body diagrams. This does not come at the expense of problem-solving accuracy. Student feedback on the activities is overwhelmingly positive.« less

Modern 3D printing technology makes it relatively easy and affordable to produce physical models that offer learners concrete representations of otherwise abstract concepts and representations. We hypothesize that integrating hands-on learning with these models into traditionally lecture-dominant courses may help learners develop representational competence, the ability to interpret, switch between, and appropriately use multiple representations of a concept as appropriate for learning, communication and analysis. This approach also offers potential to mitigate difficulties that learners with lower spatial abilities may encounter in STEM courses. Spatial thinking connects to representational competence in that internal mental representations (i.e. visualizations) facilitate work using multiple external representations. A growing body of research indicates well-developed spatial skills are important to student success in many STEM majors, and that students can improve these skills through targeted training. This NSF-IUSE exploration and design project began in fall 2018 and features cross-disciplinary collaboration between engineering, math, and psychology faculty to develop learning activities with 3D-printed models, build the theoretical basis for how they support learning, and assess their effectiveness in the classroom. We are exploring how such models can support learners’ development of conceptual understanding and representational competence in calculus and engineering statics. We are also exploring howmore »to leverage the model-based activities to embed spatial skills training into these courses. The project is addressing these questions through parallel work piloting model-based learning activities in the classroom and by investigating specific attributes of the activities in lab studies and focus groups. To date we have developed and piloted a mature suite of activities covering a variety of topics for both calculus and statics. Class observations and complementary studies in the psychology lab are helping us develop a theoretical framework for using the models in instruction. Close observation of how students use the models to solve problems and as communication tools helps identify effective design elements. We are administering two spatial skills assessments as pre/post instruments: the Purdue Spatial Visualizations Test: Rotations (PSVT:R) in calculus; and the Mental Cutting Test (MCT) in statics. We are also developing strategies and refining approaches for assessing representational competence in both subject areas. Moving forward we will be using these assessments in intervention and control sections of both courses to assess the effectiveness of the models for all learners and subgroups of learners.« less

In teaching mechanics, we use multiple representations of vectors to develop concepts and analysis techniques. These representations include pictorials, diagrams, symbols, numbers and narrative language. Through years of study as students, researchers, and teachers, we develop a fluency rooted in a deep conceptual understanding of what each representation communicates. Many novice learners, however, struggle to gain such understanding and rely on superficial mimicry of the problem solving procedures we demonstrate in examples. The term representational competence refers to the ability to interpret, switch between, and use multiple representations of a concept as appropriate for learning, communication and analysis. In engineering statics, an understanding of what each vector representation communicates and how to use different representations in problem solving is important to the development of both conceptual and procedural knowledge. Science education literature identifies representational competence as a marker of true conceptual understanding. This paper presents development work for a new assessment instrument designed to measure representational competence with vectors in an engineering mechanics context. We developed the assessment over two successive terms in statics courses at a community college, a medium-sized regional university, and a large state university. We started with twelve multiple-choice questions that survey the vector representations commonlymore »employed in statics. Each question requires the student to interpret and/or use two or more different representations of vectors and requires no calculation beyond single digit integer arithmetic. Distractor answer choices include common student mistakes and misconceptions drawn from the literature and from our teaching experience. We piloted these twelve questions as a timed section of the first exam in fall 2018 statics courses at both Whatcom Community College (WCC) and Western Washington University. Analysis of students’ unprompted use of vector representations on the open-ended problem-solving section of the same exam provides evidence of the assessment’s validity as a measurement instrument for representational competence. We found a positive correlation between students’ accurate and effective use of representations and their score on the multiple choice test. We gathered additional validity evidence by reviewing student responses on an exam wrapper reflection. We used item difficulty and item discrimination scores (point-biserial correlation) to eliminate two questions and revised the remaining questions to improve clarity and discriminatory power. We administered the revised version in two contexts: (1) again as part of the first exam in the winter 2019 Statics course at WCC, and (2) as an extra credit opportunity for statics students at Utah State University. This paper includes sample questions from the assessment to illustrate the approach. The full assessment is available to interested instructors and researchers through an online tool.« less

Perusal of any common statics textbook will reveal a reference table of standard supports in the section introducing rigid body equilibrium analysis. Most statics students eventually memorize a heuristic approach to drawing a free-body diagram based on applying the information in this table. First, identify the entry in the table that matches the schematic representation of a connection. Then draw the corresponding force and/or couple moment vectors on the isolated body according to their positive sign conventions. Multiple studies have noted how even high performing students tend to rely on this heuristic rather than conceptual reasoning. Many students struggle when faced with a new engineering connection that does not match an entry in the supports table. In this paper, we describe an inquiry-based approach to introducing support models and free body diagrams of rigid bodies. In a series of collaborative learning activities, students practice reasoning through the force interactions at example connections such as a bolted flange or a hinge by considering how the support resists translation and rotation in each direction. Each team works with the aid of a physical model to analyze how changes in the applied loads affect the reaction components. A second model of the isolatedmore »body provides opportunity to develop a tactile feel for the reaction forces. We emphasize predicting the direction of each reaction component, rather than following a standard sign convention, to provide opportunities for students to practice conceptual application of equilibrium conditions. Students’ also draw detailed diagrams of the force interactions at the mating surfaces in the connection, including distributed loadings when appropriate. We use equivalent systems concepts to relate these detailed force diagrams to conventional reaction components. Targeted assessments explore whether the approach described above might improve learning outcomes and influence how students think about free-body diagrams. Students use an online tool to attempt two multiple-choice concept questions after each activity. The questions represent near and far transfer applications of the concepts emphasized and prompt students for written explanation. Our analysis of the students’ explanations indicates that most students engage in the conceptual reasoning we encourage, though reasoning errors are common. Analysis of final exam work and comparison to an earlier term in which we used a more conventional approach indicate a majority of students incorporate conceptual reasoning practice into their approach to free-body diagrams. This does not come at the expense of problem-solving accuracy. Student feedback on the activities is overwhelmingly positive.« less